Is there a way to save a material's preview?
Short answer: no.

This preview panel really isn't meant to give you an accurate preview. It's supposed to be an "as fast as possible" preview of your material in a neutral environment. Allowing you to see what you're editing on the fly with as less waiting as possible, or scroll through your material list.
It gets refreshed at every material setting change, as well as every new selection (because you change the object, the material window gets entirely refreshed, and so does its preview panel).

The only way it could remain still when changing the selection is if Blender had an hypershade-like editor. Which will let you edit any material you want, even several at the same time, regardless of your selection. But that's not planned for now.

But to be honest, if you're working with cycles and want to have a fair preview of your object which is independent of your selection, your best shot is the 3D viewport. Because the final look of your shader is highly driven by the model's volume, the lighting, the surrounding surfaces, and their own materials, ...
If you have performance issues, there are some work around, like using the render border, using your GPU (or better if you have several GPU, use the one not used for your screen), reducing the pixel size, ...